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May 9, 2017 - Collisional Intermolecular Energy Transfer from a N2 Bath at Room Temperature to a Vibrationlly “Cold” C6F6 Molecule Using Chemical ...
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Collisional Intermolecular Energy Transfer from a N2 Bath at Room Temperature to a Vibrationlly “Cold” C6F6 Molecule Using Chemical Dynamics Simulations Amit K. Paul,†,‡ Diego Donzis,§ and William L. Hase*,† †

Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409, United States Department of Chemistry, National Institute of Technology, Meghalaya, Shillong, Meghalaya 793003, India § Department of Chemistry, Texas A&M University, College Station, Texas 77842, United States ‡

ABSTRACT: Chemical dynamics simulations were performed to study collisional intermolecular energy transfer from a thermalized N2 bath at 300 K to vibrationally “cold” C6F6. The vibrational temperature of C6F6 is taken as 50 K, which corresponds to a classical vibrational energy of 2.98 kcal/mol. The temperature ratio between C6F6 and the bath is 1/6, the reciprocal of the same ratio for previous “hot” C6F6 simulations (J. Chem. Phys. 2014, 140, 194103). Simulations were also done for a C6F6 vibrational temperature of 0 K. The average energy of C6F6 versus time is well fit by a biexponential function which gives a slightly larger short time rate component, k1, but a four times smaller long time rate component, k2, compared to those obtained from the “hot” C6F6 simulations. The average energy transferred per collision depends on the difference between the average energy of C6F6 and the final C6F6 energy after equilibration with the bath, but not on the temperature ratio of C6F6 and the bath. The translational and rotational degrees of freedom of the N2 bath transfer their energies to the vibrational degrees of freedom of C6F6. The energies of the N2 vibrational mode and translational and rotational modes of C6F6 remain unchanged during the energy transfer. It is also found that the energy distribution of C6F6 broadens as energy is transferred from the bath, with an almost linear increase in the deviation of the C6F6 energies from the average C6F6 energy as the average energy of C6F6 increases.

I. INTRODUCTION

Of interest and not well studied is energy transfer from a thermalized bath to a vibrationally “cold” molecule. This will be a counter measurement of energy transfer properties with an opposite direction of energy flow as compared to energy transfer from a vibrationally “hot” molecule to the bath. A direct measurement of the average energy transfer per collision ⟨ΔEcup⟩ can be obtained and compared with its counterpart, i.e., ⟨ΔEcdown⟩ as measured previously.33,34,43 It is also important to identify critical factors for the energy transfer such as energy and temperature differences between the seed molecule and the bath, as well as the difference in the energy/ temperature of the seed molecule and its energy/temperature after complete equilibration with the bath. The role of translational, rotational, and vibrational degrees of freedom of the seed molecule as well as the bath molecules for the energy transfer process may also be investigated. Once the initially “cold” seed molecule becomes excited, it is of interest to determine its role in energy transfer processes. Previous studies44,45 of the energy and temperature dependence for IET observed a weak temperature dependence on the average energy transfer per collision and found that it depends

Collisional intermolecular energy transfer (IET) from a highly vibrationally excited molecule (also termed as seed molecule) to a thermalized bath is important in various chemical processes, including unimolecular reaction, combustion, and atmospheric chemistry,1,2 and is well studied both experimentally3,4 and theoretically.5,6 Various experimental studies of IET from highly vibrationally excited molecules have been performed. Time resolved UV absorption spectroscopy (UVA) was used to study IET for SO2, CS2, cycloheptatrienes, toluene, and azulene molecules.7−11 An infrared fluorescence technique (IRF) was used to study pyrazine,12,13 benzene,14−17 and azulene18−22 with various collision partners. Resolved Fourier transformed infrared (FTIR) emission spectroscopy was used to study NO2.23 Azulene was also studied with time-resolved diode laser absorption spectroscopy. 24 Many atomistic simulations have been performed to investigate potential energy surface, collisional, and statistical properties of IET for molecules like benzene,25−30 toluene,10,31,32 hexafluorobenzene,6,32−34 and azulene.35−43 Moreover, the efficiency of energy transfer is tested in terms of the degree of vibrational excitation and the nature of the deactivating collision partner.33,34 Energy transfer properties have also been studied as a function of the size of the deactivating bath.33 © 2017 American Chemical Society

Received: January 30, 2017 Revised: April 17, 2017 Published: May 9, 2017 4049

DOI: 10.1021/acs.jpca.7b00948 J. Phys. Chem. A 2017, 121, 4049−4057

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The Journal of Physical Chemistry A approximately to T0.5 where T is the translational/rotational temperature. IET is also found to be more efficient with a decreased energy gap between “hot” vibration and “cold” translation motions of the excited molecule. Hexafluorobenzene (HFB) is an important molecule for studying IET. It has an energy transfer ability at least an order of magnitude higher than for other organic molecules of similar size. HFB has a clear photophysics which made it easy to handle in laser experiments. In our previous studies,33,34 a bath model was used to study IET for collisions of N2 with highly vibrationally excited HFB. The average energy of HFB, ⟨E⟩, was determined versus time, t. A sufficiently large bath of 1000 N2 molecules prevents heating of the bath, and the bath remains at a nearly constant temperature during the IET process. The slope of the resulting ⟨E⟩ versus t, i.e., d⟨E⟩/dt, when divided by the collision frequency ω, gives the average energy transferred per collision, ⟨ΔEc⟩. ⟨ΔEc⟩ is written as P(ΔEdown; E) × ⟨ΔEcdown⟩ + P(ΔEup; E) × ⟨ΔEcup⟩ where P(ΔEdown; E) and P(ΔEup; E) are the probabilities of “down” and up” energy transfer, respectively, and with the property P(ΔEdown; E) + P(ΔEup; E) = 1. ⟨ΔEcdown⟩ is the energy transfer from HFB, and ⟨ΔEcup⟩ is the energy transfer to HFB. For the C6F6 + N2 simulations with vibrationally “hot” HFB, where P(ΔEdown; E) dominates over P(ΔEup; E), excellent agreement was found between ⟨ΔEc⟩ and that determined experimentally.46 In the work reported here, the same bath model with 1000 N2 molecules is used to prevent a temperature change of the bath, and a vibrationally “cold” HFB molecule is placed at the center of the bath. This simulation is performed with a bath temperature at 300 K, the same bath temperature for the previous “hot” HFB simulations, but choosing 50 K as the vibrational temperature of the HFB. The resulting temperature ratio THFB/Tbath is 1/6, which is the reciprocal of the same ratio for the “hot” HFB simulations. An important component of this simulation study is to investigate the Landau−Teller model47 of energy transfer, which assumes same rate of energy transfer if the temperature ratio and bath temperature are kept the same. This has important implications, as this assumption has been used extensively in fundamental studies of fluid flows with a wide range of temporal and spatial scales in thermal nonequilibrium.48

(Texci/Tbath) was 6. The current simulations are motivated to see if a similar but opposite trend is followed if a ratio of 1/C is used. For the present study, including ZPE makes HFB classically “hot” and, instead of energy flow from the bath to HFB, energy flows in the opposite direction. Moreover, the current work is not to compare with experiment, but to perform the classical simulation with the same sampling technique as before, but with a “cold” instead of “hot” HFB. An initial vibrational energy of 2.98 kcal/mol was added to HFB, corresponding to a temperature of 50 K. Classical microcanonical sampling was used to add this HFB vibrational energy.49 The rotational and translational temperatures of HFB were 300 K. A sufficiently large bath was used that its temperataure was maintained at ∼300 K during the energy transfer dynamics. In the previous simulations,33,34 it was shown that the single collision limiting density is achieved for a N2 bath density of 40 kg/m3. The current simulations were done for bath densities of 20, 40, and 80 kg/m3; the latter corresponds to a pressure of 70.3 atm. For the initial HFB vibrational temperature of 50 K, and a bath of 1000 N2 molecules with a density 40 kg/m3, the temperature of the bath is only lowered by 1.3% to 294 K, upon complete thermal equilibration of HFB during the simulation. Simulations were also done at an initial HFB vibrational temperature of 0 K, i.e., classical vibrational energy of 0.0 kcal/mol. The simulations were performed using a condensed phase version of the VENUS chemical dynamics computer program.50,51 Collisional relaxation of HFB in a N2 bath was simulated by initially placing HFB at the center of a cubical box after being randomly vibrationally sampled with ∼3 or 0 kcal/ mol with translational and rotational energy sampled from Boltzmann distribution at 300 K. HFB is surrounded with N2 bath molecules thermally equilibrated. Periodic boundary conditions52 were used for the simulation box. Details of the simulation methodology have been described previously33,34 and are briefly outlined in the following. The N2 bath was equilibrated to a temperature of 300 K by a molecular dynamics (MD) simulation, as described previously.33,34,43 This equilibration is performed with the HFB molecule in the middle of the simulation cubical box, with the randomly chosen coordinates and velocities of HFB kept fixed. The MD equilibration was performed using a maximum of six stages, with each stage consisting of velocity rescaling and monitoring components of the N2 bath, for a maximum total equilibration time of 90 ps. The equilibration was complete when an accurate temperature, radial distribution function, and vibrational, rotational, and center-of-mass translational energies were obtained for the bath. After equilibration, the trajectory was further integrated for 6 ps, where seven sets of atomic positions and velocities of the N2 bath were saved at random intervals. Each of these configurations was selected as the solvent’s initial conditions for 12 trajectories, for which HFB had random initial conditions as specified by the microcanonical sampling. Each of these 7 × 12 = 84 trajectories was then further integrated by keeping HFB fixed and equilibrating the N2 bath for 12 ps. These 84 trajectories, with random initial conditions for both HFB and the N2 bath, comprised the initial trajectory ensemble. As shown previously,34 an ensemble of 24 trajectories provides semiquantitative results. All of the calculations, including the equilibrations and simulations of energy transfer, were performed with the nearest neighbor list algorithm.52 This algorithm was used to save calculation time with negligible loss in accuracy. A “cutoff”

II. POTENTIALS AND SIMULATIONS The potential energy for the C6F6 + N2 bath system is written as a sum of intramolecular and intermolecular functions V = VC6F6 + VN2 + VN2,C6F6 + VN2,N2 (1) which are defined in previous publications33,34 and are not given here. The simulations were done by considering “cold” HFB and collisional energy transfer from the N2 bath to HFB. In our previous simulations,33,34 an excitation energy of 107.4 kcal/ mol was used for HFB, which is the vibrational excitation in its ground electronic state. In that work, the importance of quasiclassical sampling, which explicitly includes zero point energy (ZPE) with the vibrational energy, was considered for the “hot” HFB simulations.34 However, the inclusion of ZPE did not have a significant effect on the simulation results.43 The classical energy of 107.4 kcal/mol without ZPE corresponds to a temperature (Texci) of 1800 K, with 30RT = 107.4 kcal/mol, where 30 is the number of C6F6 vibrational modes. The bath temperature was Tbath = 300 K. Therefore, in those simulations the ratio (C) of the temperatures of “hot” HFB and the bath 4050

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at 300 K, an average value of about 0.9 kcal/mol for both translation and rotation is added to the vibration energy, which gives a total average energy of ∼5 and ∼2 kcal/mol for the 50 and 0 K HFB vibrational temperatures, respectively. As shown in Figure 1, all the curves are well fit by the biexponential function

distance of 15 Å between the centre-of-masses of HFB and N2 was used for the nearest neighbour list. For the previous C6F6 + N2 simulations, the single collision limit was attained at a bath density of 40 kg/m3, i.e., 35 atm.33,34 The chemical dynamics simulations were performed for the three bath densities of 20, 40, and 80 kg/m 3 . The corresponding sides of the “box” for these simulations are 132.8, 105.4, and 83.7 Å. For the densities of 80, 40, and 20 kg/ m3 the trajectories were integrated up to 182, 312, and 472 ps, respectively, with a 0.004 ps of integration step size, using a sixth order symplectic numerical integration algorithm.53 These times lead to a final total HFB energy of ∼14 kcal/mol. With complete HFB thermal equilibration at 300 K, the HFB classical vibrational energy is 18 kcal/mol according to the equipartition principle of energy. As described above, an ensemble of 84 trajectories was calculated for each simulation.

⟨E(t )⟩ = [E(∞) − E(0)](1 − f1 exp(− k1t ) − f2 exp(− k 2t )) + E(0)

(2)

with f1 + f 2 = 1. E(0) and E(∞) are the initial and final energy of HFB. k1 and k2 are the rate constants. The fitting parameters are given in Table 1 for both the 50 and 0 K simulations. It is evident from Table 1 that the parameters C1 and C2 are nearly identical for the bath densities of 20 and 40 kg/m3. B. Comparison with “Hot” C6F6 Simulation. As described above, it is of interest to determine if the relaxation rate constants for “hot” HFB → bath and bath → “cold” HFB are the same, if the temperature ratio with respect to the direction of the energy flow is kept same and the bath temperature is the same. The biexponential fitting parameters for eq 2 are compared in Table 2 for the “hot” and “cold”

III. RESULTS AND DISCUSSION A. Average C6F6 Energy versus Time. In Figure 1, the average total energy of HFB, averaged over 84 trajectories, for

Table 2. Comparison of ⟨E(t)⟩ Fitting Parameters for the Hot and Cold HFB Simulationsa parameter E(∞) f1 f2 k1 k2

hot

cold

20.0 0.240 0.760 0.0138 0.00409

20.6 0.143 0.857 0.0155 0.00127

The simulations are for a density ρ = 20 kg/m3. E(∞) is in kcal/mol, and k1 and k2 are in ps−1. a

simulations for the bath density of 20 kg/m3. There are two significant differences between the parameters for the two simulations. The fraction for the short time component, f1, is 1.7 times smaller for the “cold” simulation. Though the relaxation short time component k1 is slightly larger for the “cold” simulation, the long time component k2 is close to four times smaller for the “cold” simulation. Due to this smaller k2 value, the t = 0 rate of energy transfer is almost 10 times smaller for “cold” HFB as compared to “hot” HFB. The simulations show that the rate of energy transfer depends on the energy difference between the initial HFB energy and the HFB energy after equilibration with the bath. The 300 K equilibrated HFB energy, which includes vibration, translation, and rotation energies, is ∼20 kcal/mol for both the “hot” and “cold” HFB simulations. The initial vibration energy for “hot” HFB is 107.4 kcal/mol, whereas for “cold” HFB it is

Figure 1. Average energy of HFB versus time for the bath densities of 20, 40, and 80 kg/m3. The initial vibrational energy of HFB, i.e., 2.98 kcal/mol for 50 K (top) and 0.00 kcal/mol for 0 K (bottom), was sampled from a classical microcanonical ensemble. The translational and rotational energies of HFB are taken from their distributions at 300 K, and both of their average values are ∼0.9 kcal/mol. The fit is to eq 2. There are 1000 N2 molecules in the bath.

the bath densities of 20, 40, and 80 kg/m3 is presented for two cases. In one (top), the initial HFB vibrational temperature was taken as 50 K (∼3.0 kcal/mol), whereas in the second (bottom), there is no initial HFB vibrational energy and the vibrational temperature is 0 K. Since the translational and rotational energies are sampled from a Boltzmann distribution

Table 1. Parameters for Fits to ⟨E(t)⟩a for Bath Densities of 20, 40, and 80 kg/m3 ρ (kg/m3)

E(∞)

f1

f2

k1

k2

C1

C2

0.00155 0.00310 0.00580

0.000800 0.000800 0.000787

0.0000775 0.0000775 0.0000725

0.00127 0.00258 0.00482

0.000775 0.000797 0.000824

0.0000635 0.0000645 0.0000602

0K

a

20 40 80

19.2 17.8 15.8

0.0997 0.112 0.147

0.900 0.888 0.853

20 40 80

20.6 20.0 17.6

0.143 0.139 0.177

0.857 0.861 0.823

0.0160 0.0320 0.0630 50 K 0.0155 0.0319 0.0659

The fits are to eq 2 with f1 + f 2 = 1. The k’s are in unit of ps−1. C1 and C2 are calculated by k1/ρ and k2/ρ, respectively. 4051

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The Journal of Physical Chemistry A 2.98 kcal/mol. For both simulations the initial HFB rotation and translation energies are sampled from their 300 K Boltzmann distributions, giving an average energy of 0.9 kcal/ mol for each. Thus, “hot” HFB has an initial total energy of 109.2 kcal/mol, while for “cold” HFB it is 4.8 kcal/mol. Therefore, the absolute energy difference between the initial and equilibrated average total energies are 89 and 15 kcal/mol for the “hot” and “cold” simulations, respectively. Accordingly, it is of interest to compare the rate of energy transfer for a “hot” HFB simulation having an initial HFB energy of 35 kcal/mol, with an energy difference of 15 kcal/mol with its equilibrated energy. This may be compared in the current study in terms of average energy transfer per collision, as given in the next section. In section III.E.2 below, the change in the energy distribution of HFB, P(E), during the collision energy transfer is compared for the “hot” and “cold” simulations. C. Average Energy Transfer per Collision. As was discussed in earlier studies,33,34,43 in the single collision limit, the average energy transfer per collision, ⟨ΔEc⟩, may be calculated according to ⟨ΔEc⟩ =

d⟨E⟩ 1 × dt ω

Figure 2. Average energy transferred per collision, ⟨ΔEc⟩, for the simulations using an initial vibrational energy of HFB of 2.98 kcal/mol (top) and 0.00 kcal/mol (bottom). The simulation conditions are the same as for Figure 1.

(3)

good agreement with the current ⟨ΔEc⟩ value of 0.25 kcal/mol for the “cold” HFB simulation. This indicates that the rate of energy transfer depends on the difference between the initial and equilibrated HFB energies, and not on the temperature ratio between the hot/cold HFB molecule and the bath. D. Energy Transfer Pathways. From previous work,33,43 it was found that the excitation energy of vibrationally “hot” HFB was transferred to translational and rotational energy of the N2 molecules in the bath. For the current simulation, it is of interest to see which degrees of freedom of HFB are excited by energy transfer from the bath and which degrees of freedom of the bath transfer energy to HFB. The HFB vibrational density of states, ρ(Evib), is much higher for the previous initial “hot” than the current initial “cold” HFB simulation. This could possibly affect couplings between the HFB vibrational, rotational, and translational modes during the collisional energy transfer; however, as discussed in the following, this is not the case. In Figure 3, the total translational, rotational, and vibrational energies of the bath, averaged over the 84 trajectories of the simulation, are presented versus time where HFB has an initial 50 K vibrational temperature and the bath density is 40 kg/m3. As also found in previous studies,33,43 there is no significant vibrational relaxation of the N2 bath, whereas the translational and rotational modes are relaxed in accord with the equipartition principle, i.e., keeping the ratio of the average translational and rotational energies at 1.5 throughout the dynamics. According to the equipartition principle of energy, the translational energy at 298 K of the 1000 N2 molecules is 888 kcal/mol, whereas their rotational and vibrational (potential + kinetic) energies are each 592 kcal/mol. The t = 0 initial energies of the top and the middle panel of Figure 3 are approximately 886 and 592 kcal/mol, very close to the above desired values. These energies, at 300 ps, end at 881 and 588 kcal/mol, respectively, keeping the ratio at ∼1.5. There is no significant increase or decrease in the vibrational energy of the bath throughout the dynamics. It was explained previously34 that intermolecular collisional energy transfer from an energized polyatomic molecule to cold N2 is expected to occur via low-frequency modes of the hot molecule.54,55 Much less efficient energy transfer was found for a simulation with the

where ω is collision frequency and is energy transfer per unit time. The experimental collision frequency has been given previously33 and is obtained from dE dt

⎛ T ⎞−1/2 ω = 4.415 × 107 × σ 2p⎜ ⎟ × Ω*22 ⎝ μ⎠

(4)

where σ is the collision diameter in Å, p is the pressure (Torr), T is the temperature (K), μ is in atomic mass units, and Ω22 * is collision integral taken as Ω*22 =

A C E + + DT * FT * (T *)B e e

(5)

with T* = kBT/ε, kB the Boltzmann constant, and ε is the intermolecular well depth. The constants are A = 1.16145, B = 0.14874, C = 0.52487, D = 0.77320, E = 2.16178, and F = 2.43787, with σ and ε obtained as σ = (σC6F6 + σN2)/2 and ε = εC6F6εN2 , and their values are 4.97 Å and 163 K, respectively. From the ⟨E(t)⟩ in Figure 1, with parameters in Table 1, the average energy transfer per collosion (⟨ΔEc⟩) is obtained. This data is plotted in Figure 2 for the bath densities, ρ, of 20, 40, and 80 kg/m3 and for HFB initial vibrational temperatures of 50 and 0 K. For ρ of 20 and 40 kg/m3, the ⟨ΔEc⟩ versus ⟨E⟩ plots are nearly identical for 50 K and similar for 0 K. However, there are differences between the plots for ρ of 40 and 80 kg/ m3. This observation, along with the C1 and C2 parameters for the different densities in Table 1, shows that the single collision limit for C6F6 + N2 is attained at ρ of 40 kg/m3 as found previously.33,34 ⟨ΔEc⟩ is 0.25 ± 0.01 kcal/mol for ⟨E⟩ of 5.0 kcal/mol, i.e., at the vibrational energy of ∼3.0 kcal/mol for the 50 K simulation, and considerably smaller than the value of 1.74 kcal/mol for the previous “hot” HFB simulation, with an initial vibration excitation energy of 107.4 kcal/mol and energy flow in the opposite direction. On the other hand, from the ⟨ΔEc⟩ versus ⟨E⟩ plot for the “hot” HFB simulation, the value of ⟨ΔEc⟩ is 0.30 kcal/mol for ⟨E⟩ of 35 kcal/mol, and ∼15 kcal/mol different from the equilibrated HFB energy. This value is in 4052

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∼0.9 kcal/mol at 300 K. There is no significant V ↔ T or V ↔ R energy transfer for HFB, as the bath heats “cold” HFB. It is possible that V ↔ T/R energy transfer does occur for HFB, but it is much slower than heating of HFB vibration by the N2 bath. E. HFB Energy Distribution versus Time. For the “hot” HFB simulations, the HFB energy was narrowly distributed at t = 0 and initially broadened with time. After some time, the distribution narrowed with a final P(E) very close to the Boltzmann distribution corresponding to the final temperature. It is of interest to see if there is similar behavior in the current “cold” HFB simulations, where HFB receives vibrational energy. At t = 0, the 50 K ∼3 kcal/mol vibrational energy of HFB is distributed among its vibrational normal modes using classical microcanonical normal mode sampling,45,49 with the HFB translational and rotational energies taken from their 300 K Boltzmann distributions. There is initially a narrow HFB energy distribution P(E) for the 84 trajectories. In this section, two different aspects of the variation of the HFB P(E) with time are investigated and also compared with the timedependent P(E) for the previous “hot” HFB simulations. Both the first and second moments of P(E) are considered.58−60 1. P(E) of C6F6 versus ⟨E⟩. The HFB P(E) for the “cold” simulations is analyzed at three different times. At each time HFB has an average energy ⟨E(t)⟩. At the t = 0 beginning of the simulation, the energy distribution is due to the Boltzmann distribution of the HFB initial translational and rotational energies. If only the initial HFB vibrational energy was considered, P(E) would be a delta function. In Figure 5,

Figure 3. Average energy components, center of mass translational, rotational, and vibrational (kinetic + potential) energy versus time for the simulation with bath density of 40 kg/m3 with 1000 N2 molecules in the bath. The translational and rotational energies are decreased with time, but the vibrational energy is oscillating around its initial value at 300 K. This result is from classical microcanonical sampling of initial HFB vibrational energy of 2.98 kcal/mol.

frequencies of HFB increased by changing the mass of the F atom to that of the H atom.34 Because of its very high frequency, the N2 vibration remains inactive during the collisional energy transfer. It is possible to have V ↔ T and V ↔ R energy transfer within a molecule during the collisional energy transfer,56 and intramolecular V ↔ R without a collision as a result of Coriolis coupling.57 In Figure 4, the translational, rotational, and vibrational energies of “cold” HFB are presented as a function of time. It is clear from this figure that it is only the vibrational degrees of freedom of HFB which are excited during the simulation, whereas the translational and rotational energies of HFB are fluctuating around their 3RT/2 equilibrium value of

Figure 5. Probability distribution of the HFB energy, for 84 trajectories, at the three trajectory integration times of 0, 45, and 186 ps. The average HFB energies, over the 84 trajectories, for these times are 5.10, 7.71, and 12.14 kcal/mol, respectively. Simulation results are for a N2 bath density of 40 kg/m3 and 1000 molecules in the bath.

histogram plots of P(E) are presented for the three different times. The top panel displays P(E) at t = 0, where ⟨E⟩ = 5.1 kcal/mol and the width of the energy distribution is ∼5 kcal/ mol. The middle and bottom graphs present P(E) for the two additional times of 45 and 186 ps, where ⟨E⟩ equals 7.7 and 12.1 kcal/mol, respectively. The breadth of the energy distribution is largest at 186 ps, where the width at halfmaximum is 10.0 kcal/mol compared to 3.0 and 7.0 kcal/mol

Figure 4. Same as Figure 3, but for HFB. The translational and rotational energies stay at their average value at 300 K, which is ∼0.9 kcal/mol through the simulations, whereas the energy from the bath is solely transferred to the vibrational energy of the HFB. Initial vibrational energy is 2.98 kcal/mol. 4053

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gave a value of 2.72 kcal/mol for a final temperature of 315 K for the “hot” simulation. For the current final “cold” simulation temperature of 294 K, this value is 2.60 kcal/mol. The translational and rotational contributions to ⟨ΔE2⟩1/2 may increase this value by 0.3 kcal/mol, and the resulting ⟨ΔE2⟩1/2 is in very close agreement with the value obtained from the simulation and presented in Figure 6.

for the respective times of 0 and 45.0 ps. Thus, unlike the “hot” HFB simulation where P(E) first broadens and then narrows, here the breadth of P(E) continuously broadens as HFB is vibrationally excited by the energy transfer from the bath. For the “hot” HFB simulations, the broadening and narrowing of P(E) has an insignificant effect on the average energy transfer.35 2. ⟨ΔE2⟩1/2 of C6F6 versus ⟨E(t)⟩. To obtain a clear understanding of P(E), which broadens continuously with time, the root-mean-square deviation of E from the average

(

IV. SUMMARY In this article, energy transfer from a thermalized 300 K N2 bath to vibrationally “cold” hexafluorobenzene (HFB) is considered in a classical chemical dynamics simulation. Of interest is to investigate ⟨ΔEcup⟩ as a counter to ⟨ΔEcdown⟩ calculated previously using “hot” HFB, keeping the temperature of the bath the same at 300 K. The temperature ratio of “cold” HFB and the bath, i.e., Tcold/Tbath, is taken as the reciprocal of the temperature ratio of the “hot” HFB and the bath, i.e., Thot/Tbath = 6, in the previous simulation. This is to compare energy transfer in two opposite directions of energy flow and having the same temperature ratio. In this simulation the initial vibrational energy of HFB is for a classical vibrational temperature of 50 K. This gives the temperature ratio with the bath as 1/6, the reciprocal of the previous temperature ratio of “hot” HFB and the bath.33,34 The “cold” HFB and N2 bath simulation was performed with a condensed phase version of the VENUS46,47 chemical dynamics computer program, which allows variation of the bath density. As found from the previous “hot” HFB simulations, the single collision limiting bath density is 40 kg/m3 for C6F6 + N2 IET. The current simulations were done for N2 bath densities of 20, 40, and 80 kg/m3, i.e., pressures ranging from 18 to 70 atm. The average energy ⟨E(t)⟩ of the ensemble of HFB molecules versus time, for both the previous “hot” and current “cold” simulation, is well fit by the biexponential function in eq 2. The fitting parameters for the current simulation are listed in Table 1, and the fitting parameters for the “cold” and “hot” simulations are compared in Table 2 for a bath density of 20 kg/m3. There are two substantive differences between the latter two sets of parameters; i.e., f1 is smaller and f 2 larger for the “cold” simulation, and the longer time relaxation constant k2 is ∼4 times smaller for the “cold” simulation. The average energy transferred from the bath to “cold” HFB per collision is ⟨ΔEcup⟩ = 0.25 ± 0.01 kcal/mol for the initial HFB temperature of 50 K. The HFB/bath temperature ratio for this simulation is 1/6. Obviously, this ⟨ΔEcup⟩ is opposite in sign with respect to ⟨ΔEcdown⟩ in the “hot” simulation with a temperature of 1800 K for HFB, for which the HFB/bath temperature ratio is 6. Moreover, ⟨ΔEcup⟩ obtained here is much smaller than the ⟨ΔEcdown⟩ = 1.74 kcal/mol obtained in the “hot” HFB simulation. However, if one considers the absolute magnitude of the difference between the initial HFB energy at t = 0 and the final equilibrated HFB energy in the N2 bath at 300 K, ⟨ΔEcup⟩ and ⟨ΔEcdown⟩ are comparable. The initial HFB energy for the current “cold” simulation is ∼5.0 kcal/mol, whereas the equilibrated HFB energy is ∼20 kcal/ mol. Therefore, the absolute value of the energy difference is 15 kcal/mol and ⟨ΔEcup⟩ of 0.25 ± 0.01 kcal/mol. The equilibrated energy of HFB in the simulation of “hot” HFB is the same and ∼20 kcal/mol. Consider an absolute energy difference of 15 kcal/mol for the “hot” simulation so that the average HFB energy is 35 kcal/mol. At this energy ⟨ΔEcdown⟩ is 0.30 kcal/mol and very close to the 0.25 kcal/mol obtained in

1/2

1 N

)

∑i (Ei − ⟨E⟩)2 value, i.e., ⟨ΔE2⟩1/2 = , was calculated. In 2 1/2 Figure 6, ⟨ΔE ⟩ is plotted versus ⟨E(t)⟩. Compared to the

Figure 6. Root-mean-square deviation of the trajectories’ total energy with respect to the average HFB energy,

⟨ΔE2⟩1/2 =

1 N

∑i (Ei − ⟨E⟩)2 , N = 84. Ei is the energy of the ith

trajectory. Simulation results are for a N2 bath density of 40 kg/m3 and 1000 molecules in the bath.

“hot” HFB simulations, for which ⟨ΔE2⟩1/2 first increases and then decreases,34 here ⟨ΔE2⟩1/2 increases almost linearly up to the final ⟨E(t)⟩. This may be understood by considering the different energy transfer dynamics for the “hot” and “cold” simulations. For the “hot” HFB simulation, the initial monoenergetic vibrational energy is equivalent to a temperature of 1800 K, with the initial P(E) quite narrow resulting from its 300 K translation and rotation energy Boltzmann distribution. As the trajectories are integrated, P(E) broadens, reflecting the high HFB internal energy and an equivalent high temperature. At longer times the distribution then narrows to attain the 300 K thermalized distribution. This is not the case for the “cold” simulation, where the P(E) only broadens from its narrow initial distribution to the thermalized 300 K distribution. Unlike the “hot” simulation, there is not an initial high internal energy/ temperature first driving the broadening of P(E). This is consistent with the P(E) obtained for the current “cold” simulations, with P(E) broadening until equilibration and the final HFB temperature. The following analytical expression for the vibrational P(Evib) was considered and evaluated for the previous “hot” HFB simulation, i.e.,

( ) ) exp(− ) E

P(Evib) =

ρ(Evib) exp − k vibT B

∞

∫0 ρ(Evib

Evib kBT

(6)

Using a Monte Carlo technique, where ρ(Evib) is the vibrational density of states, ⟨ΔE2⟩1/2 was calculated for this P(Evib). This 4054

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The Journal of Physical Chemistry A

High Performance Computing Center (HPCC) at Texas Tech University, under the direction of Philip W. Smith. Parts of the computations were also performed on Robinson, a general computer cluster of the Department of Chemistry and Biochemistry, Texas Tech University, purchased by the NSF CRIF-MU grant CHE-0840493.

the present simulation. Therefore, it may be concluded that the average value of the energy transfer per collision does not depend on the temperature ratio of the HFB and the bath, but their energy difference. The energy transfer from the bath is primarily from the translational and rotational degrees of freedom of the N2 molecules, with negligible energy change in N2 vibration. This transfer of bath translational and rotational energies is consistent with the equipartition principle, i.e., the ratio of the bath translational and rotational energies is maintained at 1.5 throughout the dynamics. The energy transfer from the bath is used to excite the HFB vibrational degree of freedom, whereas the HFB translational and rotational energies fluctuate around their thermal equilibrium value of 0.9 kcal/mol. V → T or V → R energy transfer for the HFB degrees of freedom is unimportant. The energy distribution P(E) of HFB was also studied. For the previous “hot” HFB simulation,34 P(E) first broadened and then narrowed as HFB was equilibrated with the bath. For the current “cold” HFB simulation, P(E) continuously broadens as energy is transferred to HFB and it equilibrates with the bath. This increasing width of P(E) is seen by calculating the rootmean-square deviation from the average ⟨E(t)⟩. The result is presented in Figure 6, where an almost linear increase in this deviation is found as a function of the average HFB energy. There are multiple possible extensions of this and previous studies33,34,43 of collisional intermolecular energy transfer (IET) using the condensed phase software described here. It is important to consider different hot/cold and bath molecules to acquire information regarding molecular properties and IET, which may be used to develop theoretical models which describe the IET. It is important to consider polyatomic bath molecules, with a range of vibrational frequencies, to begin to quantify the role of bath vibrational modes for the energy transfer.20 Studying IET at high densities and in the liquid phase will be important,61 as will comparisons with experiment. As discussed previously,62 classical chemical dynamics simulations provide the rate of energy transfer and the energy transfer collision frequency is needed to determine the average energy transfer per collision. For the comparisons made here with experiment, the experimental energy transfer collision frequency is used. In future work, it may be of interest to investigate the possibility of obtaining the energy transfer collision frequency directly from the simulations. Different models have been proposed for doing this,63,64 and it would be of interest to establish if there are models which recover the experimental energy transfer collision frequency.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Amit K. Paul: 0000-0001-7074-1232 William L. Hase: 0000-0002-0560-5100 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research reported here is based upon work supported by the Air Force Office of Scientific Research (AFOSR) BRI grant FA 9550-12-1-0443 and the Robert A. Welch Foundation under Grant No. D-0005. Support was also provided by the 4055

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